Integrand size = 19, antiderivative size = 76 \[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=-\frac {2 a \sqrt {-a+b (c x)^n}}{n}+\frac {2 \left (-a+b (c x)^n\right )^{3/2}}{3 n}+\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )}{n} \]
2/3*(-a+b*(c*x)^n)^(3/2)/n+2*a^(3/2)*arctan((-a+b*(c*x)^n)^(1/2)/a^(1/2))/ n-2*a*(-a+b*(c*x)^n)^(1/2)/n
Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\frac {-2 \left (4 a-b (c x)^n\right ) \sqrt {-a+b (c x)^n}+6 a^{3/2} \arctan \left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )}{3 n} \]
(-2*(4*a - b*(c*x)^n)*Sqrt[-a + b*(c*x)^n] + 6*a^(3/2)*ArcTan[Sqrt[-a + b* (c*x)^n]/Sqrt[a]])/(3*n)
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {891, 27, 798, 60, 60, 73, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (b (c x)^n-a\right )^{3/2}}{x} \, dx\) |
\(\Big \downarrow \) 891 |
\(\displaystyle \frac {\int \frac {\left (b (c x)^n-a\right )^{3/2}}{x}d(c x)}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\left (b (c x)^n-a\right )^{3/2}}{c x}d(c x)\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \frac {\left (b (c x)^n-a\right )^{3/2}}{c x}d(c x)^n}{n}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {\frac {2}{3} \left (b (c x)^n-a\right )^{3/2}-a \int \frac {\sqrt {b (c x)^n-a}}{c x}d(c x)^n}{n}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {\frac {2}{3} \left (b (c x)^n-a\right )^{3/2}-a \left (2 \sqrt {b (c x)^n-a}-a \int \frac {1}{c x \sqrt {b (c x)^n-a}}d(c x)^n\right )}{n}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {2}{3} \left (b (c x)^n-a\right )^{3/2}-a \left (2 \sqrt {b (c x)^n-a}-\frac {2 a \int \frac {1}{\frac {c^2 x^2}{b}+\frac {a}{b}}d\sqrt {b (c x)^n-a}}{b}\right )}{n}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {2}{3} \left (b (c x)^n-a\right )^{3/2}-a \left (2 \sqrt {b (c x)^n-a}-2 \sqrt {a} \arctan \left (\frac {\sqrt {b (c x)^n-a}}{\sqrt {a}}\right )\right )}{n}\) |
((2*(-a + b*(c*x)^n)^(3/2))/3 - a*(2*Sqrt[-a + b*(c*x)^n] - 2*Sqrt[a]*ArcT an[Sqrt[-a + b*(c*x)^n]/Sqrt[a]]))/n
3.7.65.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Simp[1/c Subst[Int[(d*(x/c))^m*(a + b*x^n)^p, x], x, c*x], x] /; FreeQ[{a , b, c, d, m, n, p}, x]
Time = 1.41 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (-a +b \left (c x \right )^{n}\right )^{\frac {3}{2}}}{3}-2 a \sqrt {-a +b \left (c x \right )^{n}}+2 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n}\) | \(60\) |
default | \(\frac {\frac {2 \left (-a +b \left (c x \right )^{n}\right )^{\frac {3}{2}}}{3}-2 a \sqrt {-a +b \left (c x \right )^{n}}+2 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n}\) | \(60\) |
risch | \(\frac {2 \left (-b \,{\mathrm e}^{n \ln \left (c x \right )}+4 a \right ) \left (a -b \,{\mathrm e}^{n \ln \left (c x \right )}\right )}{3 n \sqrt {-a +b \,{\mathrm e}^{n \ln \left (c x \right )}}}+\frac {2 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a +b \,{\mathrm e}^{n \ln \left (c x \right )}}}{\sqrt {a}}\right )}{n}\) | \(76\) |
1/n*(2/3*(-a+b*(c*x)^n)^(3/2)-2*a*(-a+b*(c*x)^n)^(1/2)+2*a^(3/2)*arctan((- a+b*(c*x)^n)^(1/2)/a^(1/2)))
Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.78 \[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\left [\frac {3 \, \sqrt {-a} a \log \left (\frac {\left (c x\right )^{n} b + 2 \, \sqrt {\left (c x\right )^{n} b - a} \sqrt {-a} - 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt {\left (c x\right )^{n} b - a} {\left (\left (c x\right )^{n} b - 4 \, a\right )}}{3 \, n}, \frac {2 \, {\left (3 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {\left (c x\right )^{n} b - a}}{\sqrt {a}}\right ) + \sqrt {\left (c x\right )^{n} b - a} {\left (\left (c x\right )^{n} b - 4 \, a\right )}\right )}}{3 \, n}\right ] \]
[1/3*(3*sqrt(-a)*a*log(((c*x)^n*b + 2*sqrt((c*x)^n*b - a)*sqrt(-a) - 2*a)/ (c*x)^n) + 2*sqrt((c*x)^n*b - a)*((c*x)^n*b - 4*a))/n, 2/3*(3*a^(3/2)*arct an(sqrt((c*x)^n*b - a)/sqrt(a)) + sqrt((c*x)^n*b - a)*((c*x)^n*b - 4*a))/n ]
Time = 11.99 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.24 \[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\begin {cases} \frac {\begin {cases} 2 a^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {- a + b \left (c x\right )^{n}}}{\sqrt {a}} \right )} - 2 a \sqrt {- a + b \left (c x\right )^{n}} + \frac {2 \left (- a + b \left (c x\right )^{n}\right )^{\frac {3}{2}}}{3} & \text {for}\: b \neq 0 \\- a \sqrt {- a} \log {\left (\left (c x\right )^{n} \right )} & \text {otherwise} \end {cases}}{n} & \text {for}\: n \neq 0 \\\left (- a \sqrt {- a + b} + b \sqrt {- a + b}\right ) \log {\left (x \right )} & \text {otherwise} \end {cases} \]
Piecewise((Piecewise((2*a**(3/2)*atan(sqrt(-a + b*(c*x)**n)/sqrt(a)) - 2*a *sqrt(-a + b*(c*x)**n) + 2*(-a + b*(c*x)**n)**(3/2)/3, Ne(b, 0)), (-a*sqrt (-a)*log((c*x)**n), True))/n, Ne(n, 0)), ((-a*sqrt(-a + b) + b*sqrt(-a + b ))*log(x), True))
\[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\int { \frac {{\left (\left (c x\right )^{n} b - a\right )}^{\frac {3}{2}}}{x} \,d x } \]
\[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\int { \frac {{\left (\left (c x\right )^{n} b - a\right )}^{\frac {3}{2}}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\int \frac {{\left (b\,{\left (c\,x\right )}^n-a\right )}^{3/2}}{x} \,d x \]